Question

# Find the coefficient of ${{x}^{n}}$ in the expansion of $\left( 1+x \right){{\left( 1-x \right)}^{n}}$ ?(a) ${{\left( -1 \right)}^{n-1}}n$ (b) ${{\left( -1 \right)}^{n}}\left( 1-n \right)$ (c) ${{\left( -1 \right)}^{n-1}}{{\left( n-2 \right)}^{2}}$ (d) $\left( n-1 \right)$

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Hint: We start solving the problem by expanding the given multiplication and then find the terms for which we need to find the coefficient. We then recall the fact that the coefficient of ${{x}^{r}}$ in the binomial expansion of ${{\left( a+bx \right)}^{n}}$ , $\left( n\ge r \right)$ as ${}^{n}{{C}_{r}}{{a}^{n-r}}{{b}^{r}}$ to find the coefficients of required terms. We then add those obtained coefficients and make the necessary calculations to get the required answer.

Complete step by step answer:
According to the problem, we are asked to find the coefficient of ${{x}^{n}}$ in the expansion of $\left( 1+x \right){{\left( 1-x \right)}^{n}}$ .
Now, we have $\left( 1+x \right){{\left( 1-x \right)}^{n}}={{\left( 1-x \right)}^{n}}+x{{\left( 1-x \right)}^{n}}$ ---(1).
In order to find the coefficient of ${{x}^{n}}$ for the expansion in equation (1), We need to find the coefficient of ${{x}^{n}}$ for ${{\left( 1-x \right)}^{n}}$ and ${{x}^{n-1}}$ for ${{\left( 1-x \right)}^{n}}$ to get required coefficient in $x{{\left( 1-x \right)}^{n}}$ .
Let us recall the coefficient of ${{x}^{r}}$ in the binomial expansion of ${{\left( a+bx \right)}^{n}}$ , $\left( n\ge r \right)$ as ${}^{n}{{C}_{r}}{{a}^{n-r}}{{b}^{r}}$ .
Now, the co-efficient of ${{x}^{n}}$ in the expansion of ${{\left( 1-x \right)}^{n}}$ is ${}^{n}{{C}_{n}}{{\left( 1 \right)}^{n-n}}{{\left( -1 \right)}^{n}}$ .
We know that ${}^{n}{{C}_{n}}=1$ . So, the co-efficient of ${{x}^{n}}$ in the expansion of ${{\left( 1-x \right)}^{n}}$ is ${{\left( -1 \right)}^{n}}$ ---(2).
Now, the co-efficient of ${{x}^{n-1}}$ in the expansion of ${{\left( 1-x \right)}^{n}}$ is ${}^{n}{{C}_{n-1}}{{\left( 1 \right)}^{n-n+1}}{{\left( -1 \right)}^{n-1}}$ .
We know that ${}^{n}{{C}_{n-1}}=n$ . So, the co-efficient of ${{x}^{n-1}}$ in the expansion of ${{\left( 1-x \right)}^{n}}$ is ${{\left( -1 \right)}^{n-1}}n$ ---(3).
Let us use the results obtained in equations (2) and (3) to get the coefficient of ${{x}^{n}}$ in the expansion of $\left( 1+x \right){{\left( 1-x \right)}^{n}}$ .
So, the required co-efficient is ${{\left( -1 \right)}^{n}}+{{\left( -1 \right)}^{n-1}}n={{\left( -1 \right)}^{n}}+\dfrac{{{\left( -1 \right)}^{n}}}{\left( -1 \right)}n={{\left( -1 \right)}^{n}}\left( 1-n \right)$ .
We have found the coefficient of ${{x}^{n}}$ in the expansion of $\left( 1+x \right){{\left( 1-x \right)}^{n}}$ as ${{\left( -1 \right)}^{n}}\left( 1-n \right)$ .
$\therefore$ The correct option for the given problem is (b).

Note:
We should not confuse while finding the binomial coefficients of required terms in this problem. Whenever we get this type of problem, we first try to find the required terms whose coefficients we need to find for solving the problem. Similarly, we can expect problems to find the coefficient of ${{x}^{n}}$ in the binomial expansion of $\left( 1+x \right){{\left( 1-x \right)}^{-n}}$ .